The Computational Complexity

I would like to say that I am thrilled to take over as editor of this column. But I am more \daunted" than \thrilled". Juris Hartmanis started this column back in issue # 31 of the Bulletin, and it has been an extremely successful run. I'm not eager to invite comparison. In upcoming editions of this column, you should expect to nd concise and readable accounts of some exciting developments and trends in complexity theory. Often, this may take the form of a survey or tutorial on some body of recent work. In this issue of the Bulletin, however, I will survey some aspects of complexity theory that can hardly be considered \recent". Instead, I will focus on some fundamental truths relating complexity theory to practical computing. These truths deserve repeating, lest they become forgotten. As a bonus, this edition of the Computational Complexity Column concludes with a report by Jack Lutz on the recent sixtieth birthday celebration for Ron Book. Book has had a profound and lasting innuence on the eld of complexity theory, and I am happy to be able to include a report on such an event in this inaugural column. I am surprised at how often I encounter skepticism about the relevance of an asymp-totic lower bound on the computational complexity of a function. Admittedly, the grounds for this skepticism may initially sound plausible, as the following \straw man" argument may illustrate: Let's assume that you've shown that function f requires time 2 (n) to compute, and also requires circuit size 2 (n). This still doesn't tell me that I won't be able to compute f for my application. I'm never going to need 1 to deal with inputs with more than a billion bits, and your lower bound only says that there is some > 0 such that the circuit size is at least 2 n. But if turns out to be one-billionth, then this tells me nothing. In fact, the set I'm interested in is nite, and all nite sets are regular, and hence they're recognizable in linear time. I'm afraid that the sort of asymptotic bounds provided by complexity theory miss the boat completely! If you nd this argument convincing, please read further. It is true that lower bounds in complexity theory are usually stated in terms of asymptotic bounds (in the rare cases when actual lower bounds are able to be presented …